Question: Determine how many solutions exist for the system of equations. ${-3x-y = -1}$ ${10x+2y = 4}$
Explanation: Convert both equations to slope-intercept form: ${-3x-y = -1}$ $-3x{+3x} - y = -1{+3x}$ $-y = -1+3x$ $y = 1-3x$ ${y = -3x+1}$ ${10x+2y = 4}$ $10x{-10x} + 2y = 4{-10x}$ $2y = 4-10x$ $y = 2-5x$ ${y = -5x+2}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -3x+1}$ ${y = -5x+2}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.